Color Confinement and Hadrons

The following sections are included:

• PREFACE

• CONTENTS

• Welcome Greetings

This is the short review of Monte-Carlo studies of quark confinement in lattice QCD. After abelian projections both in the maximally abelian and Polyakov gauges, it is seen that the monopole part alone is responsible for confinement. A block spin transformation on the dual lattice suggests that lattice SU(2) QCD is always (for all ß) in the monopole condensed phase and so in the confinement phase in the infinite volume limit.

The condensation of monopoles (dual superconductivity) of QCD vacuum is reviewed. Direct evidence is produced that the system, in the confined phase, is a dual superconductor.

The spacial distribution of fields and currents in confining theories can give direct evidence of dual superconductivity. We would like to discuss the techniques for finding these properties and calculating the superconductivity parameters in lattice simulations. We have seen dual Abrikosov vortices directly in pure U(l) and SU(2) and others have also seen them in SU(3). In the non-Abelian cases the system appears to be near the borderline between type I and II. We also discuss the response of the supercurrents to external fields.

Aspects of the monopole condensation picture of confinement are discussed. First, the nature of the monopole singularities in the abelian projection approach is analysed. Their apparent gauge dependence is shown to have a natural interpretation in terms of't Hooft-Polyakov-like monopoles in euclidean SU(2) gauge theory. Next, the results and predictions of a realization of confinement through condensation of such monopoles are summarized and compared with numerical data.

First we discuss various topological objects (monopoles, “minopoles” and “hybrids”) which may be important for the confinement mechanism in various abelian projections. The second topic is the string between quark and antiquark. The standard quantum string with the Nambu-Goto action exists only in D=26. If we start from the field theory, in which the string excitations exist, and change the variables in the path integral to the string variables, then the Jacobian appears. This Jacobian generates the correction to the Nambu-Goto action. For this effective action the conformal anomaly cancels in D=4. Thus we get the quantum string theory in D=4.

We describe long distance QCD by a dual theory in which the fundamental variables are dual potentials coupled to monopole fields and use this dual theory to determine the effective Lagrangian for constituent quarks. We find the color field distribution surrounding a quark anti-quark pair to first order in their velocities. Using these distributions we eliminate the dual potentials and obtain an effective interaction Lagrangian depending only upon the quark and anti-quark coordinates and velocities, valid to second order in their velocities. We propose L_I as the Lagrangian describing the long distance interaction of constituent quarks.

Dual Ginzburg-Landau(DGL) theory is constructed assuming abelian dominance and monopole condensation in abelian projected QCD which are supported by Monte Carlo simulations in lattice QCD. The applications of DGL theory to static meson, baryon, meson-meson and finte temperature systems are discussed.

Nonperturbative QCD is studied with the dual Ginzburg-Landau theory, where color confinement is realized through the dual Higgs mechanism by QCD-monopole condensation. We obtain a general analytic formula for the string tension. A compact formula is derived for the screened inter-quark potential in the presence of light dynamical quarks. The QCD phase transition at finite temperature is studied using the effective potential formalism. The string tension and the QCD-monopole mass are largely reduced near the critical temperature, T_c. The surface tension is estimated from the effective potential at T_c. We propose also a new scenario of the quark-gluon-plasma creation through the color-electric flux-tube annihilation. Finally, we discuss a close relation between instantons and QCD-monopoles.

We analyze topological objects in pure QCD in the presence of external quarks by calculating the distributions of instanton and monopole densities around static color sources. We find a suppression of the densities close to external sources and the formation of a flux tube between a static quark-antiquark pair. The similarity in the behavior of instantons and monopoles around static sources might be due to a local correlation between these topological objects. On an 8³ × 4 lattice at β = 5.6, it turns out that topological quantities are correlated approximately two lattice spacings.

We present results on the spin-independent quark-antiquark potential in SU(3) gauge theory from a simulation on a 48³ × 64 lattice at β = 6.8, corresponding to a volume of (1.7 fm)³. Moreover, a comprehensive analysis of spin- and velocity-dependent potentials is carried out for SU(2) gauge theory, with emphasis on the short range structure, on lattices with resolutions ranging from. 02 fm to. 04 fm.

We present SU(3) gluon propagators calculated on 48 × 48 × 48 × N_t lattices at β = 6.8 where N_t = 64 (corresponding the confinement phase) and N_t = 16 (deconfinement) with the bare gauge parameter,α, set to be 0.1. In order to avoid Gribov copies, we employ the stochastic gauge fixing algorithm. Gluon propagators show quite different behavior from those of massless gauge fields: (1) In the confinement phase, G(t) shows massless behavior at small and large t, while around 5 < t < 15 it behaves as massive particle, and (2) effective mass observed in G(z) becomes larger as z increases. (3) In the deconfinement phase, G(z) shows also massive behavior but effective mass is less than in the confinement case. In all cases, slope masses are increasing functions of t or z, which can not be understood as addtional physical poles.

In this talk I discuss the current picture of color confinement and show how it can be tested microscopically. It is stressed that the color magnetic monopoles in this picture are dyons. Furthermore, the role of instantons is illuminated.

We discuss recent results from lattice studies of SU(N) gauge theories at finite temperature. We will present results from a calculation of electric and magnetic screening masses for the gluons, discuss calculations of the critical temperature in units of the string tension and bulk thermodynamic quantities like the energy density and pressure. In particular, the latter calculations have now reached a stage where O(a²) cut-off effects can be controlled systematically and an extrapolation to the continuum limit can be performed.

The effect of the strange quark in the finite temperature phase transition of QCD is studied on the lattice. Using the one-plaquette gauge action and the Wilson quark action, the transition in the chiral limit is shown to be continuous for the case of degenerate two flavors, N_F = 2, while it is of first order for N_F ≥ 3. For a more realistic case of massless up and down quarks and a light strange quark, N_F = 2 + 1, clear two state signals are observed both for m_n ≃ 150 and 400 MeV. In contrast to a previous result with staggered quarks, this suggests a first order transition in the real world. In order to see the implication of these results to the continuum limit, we started to study these issues using improved actions. First results using a RG improved gauge action combined with the standard Wilson quark is presented for the case of N_F = 2: With this action the finite temperature transition is shown to be continuous in the chiral limit confirming the result of the standard action. Furthermore, not like the case of the standard action where lattice artifacts make the transition once very strong at intermediate values of the hopping parameter K on N_t = 4 and 6 lattices, a smooth crossover is found for the improved action when we increase 1/K — l/K_c, in accord with a naive expectation about the fate of second order chiral transition at finite m_q

We discuss problems related to the analysis of finite temperature 4-point correlators in the mesonic chanels. We present preliminary results on the pion wave function and dispersion law in quenched QCD at T ≃ 0.83T_C.

The mass dependence of the chiral phase transition is studied in the linear SU(3) × SU(3) sigma model to leading order in a 1/N-expansion. The chiral transition is washed out for an average pseudoscalar meson octet mass ≥ 203 [MeV]. The corresponding ratio of critical to physical light quark condensates is estimated as ∼ 30 %, while it is only ∼ 3 % in a meanfield approximation. A ratio close to 1 would be desirable for visible effects in RHIC-experiments. For physical meson masses we find a rapid crossover in the normalized light quark condensate for 181.5 ≤ T ≤ 192.6 [MeV]. The entropy density increases by 5.5 ± 0.8 · 10⁻³ [GeV³] in the same temperature interval. Due to large errors in the transition region we cannot exclude a finite latent heat ΔL ≤ 0.2 [GeV/fm³]. This bound is only 10 % of the prediction from the bag model equation of state.

The field strength approach to Yang-Mills theory is formulated in the maximal Abelian gauge. In this way SU(2) Yang-Mills theory is transformed into an effective theory in the Abelian part of the gauge field and a tensor field, which describes the non-Abelian features. The latter is Abelian for SU(2).

It is shown that the semirelativistic potential, the relativistic flux tube model and a confining Bethe–Salpeter equation can be derived from QCD first principles in a unified point of view.

We derive a confining Bethe–Salpeter equation starting from the same assumptions on the Wilson loop integral already adopted in the derivation of a semirelativistic heavy quark potential. We show that, by standard approximations, an effective meson squared mass operator can be obtained from our BS kernel and that, from this, by expansion, the corresponding Wilson loop potential is recovered, spin-dependent and velocity-dependent terms included. We also show, that, on the contrary, neglecting spin–dependent terms, relativistic flux tube model is reproduced.

The Bethe-Salpeter (BS) equation for scalar-scalar bound states in scalar theories without derivative coupling is formulated and solved in Minkowski space. This is achieved using the perturbation theory integral representation (PTIR), which allows these amplitudes to be expressed as integrals over weight functions and known singularity structures and hence allows us to convert the BS equation into an integral equation involving weight functions. We obtain numerical solutions using this formalism for a number of scattering kernels to illustrate the generality of the approach. It applies even when the naive Wick rotation is invalid. As a check we verify, for example, that this method applied to the special case of the massive ladder exchange kernel reproduces the same results as are obtained by Wick rotation.

Behaviour of deep-inelastic structure function at lower Q² ≤ 1GeV² is discussed, with a mass scale of order m² ≃ 0.5 GeV² suggested to be universal. This value is apparently different from the known QCD mass-scale parameter, namely . It is pointed out that the larger mass scale hereby suggested is known and/or found in accordance with the other sources of information from available experimental data and attempted analyses of the data such like hadron spectroscopy, higher-twist terms, R-ratio in e⁺e⁻ reaction, and theoretical requirement for the confinement.

Recent development on the studies of the nucleon's chiral-odd spin-structure functions h₁(x, Q²) is reviewed. These spin structure functions can be identified by measuring proper asymmetries in the nucleon-nucleon polarized Drell-Yan processes. The Q² evolution of the twist-three structure function h_L(x, Q²) is described in detail.

After a brief introduction to QCD sum rules, we discuss its applications to extract longitudinal and transverse quark distribution amplitudes in mesons. For both directions, the results for the pion and the longitudinal rho meson show marked qualitative differences. These results are compared to the calculations from, Lattice QCD, the light-cone constituent quark model and the NJL model.

A criterion for dynamical chiral symmetry breaking in gauge theories is obtained in the improved ladder approximation. Especially, in gauge theories in the Higgs phase, the condition for dynamical chiral symmetry breaking is given by N_B ≥ 0.36, where N_B denotes the number of bound states estimated in the WKB approximation.

Recent developments in the improved ladder approximation are reviewed as applied to the Schwinger-Dyson and the Bethe-Salpeter equations in QCD-like theories. We explain the calculations of the pion decay constant f_π, the VEV and the mass values and decay constants of the lowest lying mesons in QCD.

We show on the basis of an effective theory of QCD that a wide variety of observables in the hadron world is governed by the chiral symmetry together with an interplay between the axial anomaly and the explicit symmetry breaking due to the current quark mass. We also discuss the nature of the chiral transition at finite temperature and related dynamical phenomena using the effective Lagrangian. Some phenomenological implications of the small vector coupling (“vector limit”) at high temperatures are suggested.

We study the properties of quarks, being confined in hadrons, with the Schwinger-Dyson equation in the dual Ginzburg-Landau Theory. Magnetic monopole condensation, which provides quark confinement, is demonstrated responsible also for dynamical chiral-symmetry breaking. We discuss then the recovery of the chiral symmetry at finite temperature.

Pseudoscalar correlator is measured in a singular (monopole dominant) and a regular (photon dominant) parts of a maximal abelian field on SU(2) lattice. In the abelian field and its singular part, light pseudoscalar boson are observed similar to that in SU(2) field. On the other hand, the correlator in the regular part behaves like a product of free quark and anti-quark. Obtained results give a support for a possibility that monopole condensation is responsible for chiral symmetry breaking as well as confinement. Correlation between monopoles and instantons is also investigated by observing topological charge. A field including monopoles carrys topological charge with good correlation to that in original SU(2) gauge field while that including only photons shows no topological charge.

We present evidence that the existence of a first order phase transition in compact U(l) with Wilson action is not related to monopole loops wrapping around the toroidal lattice, as has been previously suggested. Our analysis is based on the suppression of such loops by ‘soft boundary conditions’ that correspond to an infinitely large chemical potential for the monopoles on the boundary, during the updating process. It is observed that the double peak structure characteristic for the first order phase transition reappears at sufficiently large lattice sizes and separations from the lattice boundary.

We discuss a class of saddle-point configurations in SU(2) lattice gauge theory in three Euclidean dimensions. These configurations are smooth on the scale of the lattice and have an action density exhibiting localized peaks, as has been seen in cooled and extremized Monte Carlo generated lattices. Large Wilson loops centered on the action peaks show a unit of Z(2) flux. We discuss the generation of these configurations and measurements of the Creutz ratios on them.

A first order cosmological QCD phase transition and the creation of inhomogeneous baryon-density distribution are discussed. Physical condition on the fluctuation amplitude, length scale and the QCD parameters is theoretically studied. Impact of the QCD physics on several cosmological problems is reviewed.

The primary goals of PHENIX experiment at Relativistic Heavy Ion Collider (RHIC) are to detect a new phase of matter, Quark-Gluon Plasma (QGP) and to measure its properties. The QGP is considered to have existed in the early Universe right after the Big Bang.

A brief report is given on experimental programs relevant to quark nuclear physics at Research Center For Nuclear Physics, Osaka University. Subjects discussed are present and perspective of nuclear physics experiments at RCNP, hypernuclear spectroscopy and strange quarks, quark nuclear spectroscopy with multi-GeV photons from SPring-8, and super computer system for QCD/lattice calculations.

We study the relation between the quark confinement and the dynamical chiral symmetry breaking in SU(2) QCD by deriving an effective Lagrangian of a monopole field and the chiral fields from the dual Ginzburg-Landau type Lagrangian(DGL Lagrangian)…

We estimate the string tension performing two ways of abelian projection to investigate the gauge independence of 'tHooft conjecture in dual Meissner effect.

We investigate the role of monopole currents in the finite temperature QCD, using Monte-Carlo simulations of lattice QCD. The purpose of our study is to confirm that the dual Meissner effect due to condensation of color magnetic monopole is color confinement mechanism in QCD…

The structure of a flux tube is studied in SU(2) lattice gauge theory from the viewpoint of the abelian projection theory. The data suggest that the flux distributions of the transverse electric fields and the magnetic fields are produced by the effect that the abelian monopoles in the maximally abelian gauge are expelled from the string region, and that the longitudinal electric flux is formed from the abelian photon field and the monopole contribution.

We investigate the chiral symmetry restoration at finite temperature by solving the Schwinger-Dyson equation in the Dual Ginzburg-Landau theory, which provides both color confinement and chiral symmetry breaking.

We find deconfinement phase transition into the quark-gluon-plasma at finite temperature and then show a possibility of QGP formation process due to the interaction of color-flux-tubes in dual Ginzburg-Landau (DGL) theory.

We calculate the surface tension between two phases in the dual Ginzburg-Landau theory in two cases; one is between the quark gluon plasma and the hadron phases at finite temperature and the other in the coexistence system of two phases with the external color electric fields at zero temperature.

In three-dimensional QED (QED3), we show that there is a connection between the confining potential and complex singularities of the fermion propagator.

1+1 dimensional SU(N) quantum chromodynamics (QCD(N)₂) is a good model for studying ideas and tools which might be feasible in analyses of QCD in 3+1 dimensions. The color confinement in this model is enforced by the term

The spin-orbit component phase-shifts and the spin-orbit eikonals of p-p scattering are determined from the solutions of the phase-shift analyses of the experimental

data at 3, 6 and 12 GeV/c, respectively. Its implications in the first order phase transition in

the p-p scattering in this energy recattering in the incident momentum region P_L=l∼l2 GeV/c. An evidence of a strong repulsive spin-orbit force in the short distance b≲0.5 fm was obtained in our previous work. But our ordinary explanation about the energy dependence of spin-orbit P-wave phase shift may be open to question because of too drastic energy-dependence of new short-range force. The eikonal is numerically evaluated by Fourier-Bessel transformation of the scattering matrix M(q) of the PSA solutions. The real part of the obtained spin-orbit eikonal Re_χLS presents the information on the b–dependence of the spin–orbit interaction. The solutions of Reχ_LS at 3, 6 and 12 GeV/c agree among them in the outer region b≳1.0 fm, which represent the force due to peripheral mesonic contibution. In the region of 0.5∼1.0 fm, Reχ_LS exhibits the nonstatic effect due to the modified one-boson-exchange interaction. In the inner region of b≲0.5 fm, Reχ_LS gives new

indication on an existence of a short-range repulsive force with a strong energy dependence. Our only possible explanation of this behavior seems to be that a phase transition takes place in the incident energy region of proton T_L= 3∼10 GeV. The threshold radius for the “ shrinkage ” of the spin-orbit force is about 0.5 fm, which almost agrees with the bag-radius predicted by the MIT bag model. The critical temperature(T_c) for the transition from the quark-gluon-plasma phase to the hadron phase is estimated as 150∼200 MeV by using the Gibbs conditiic transition of the spin-orbit force.

In the SMC experiment at CERN, it was reported that the quark spin contribution to the nucleon spin is unexpectedly small. This result called “ spin crisis ” seems to be in accordance with our discovered “ shrinkage ” of the spin-dependent nuclear force.

We propose a new method ¹ for the study of the chiral properties of the ground state in QFT′s, based on the computation of the probability distribution function (p.d.f) of the chiral condensate. The advantage is that one can obtain this order parameter without introducing an external field. Here we just outline the strategy…

We derive the low energy theorem for the mass and the decay constant of the pion at finite density. The result turns out to be consistent with the calculation based on the QCD sum rule.

Assuming Coulomb-like as well as confining scalar potential, we have perturbatively solved Schödinger equation up to second order in 1/m_Q with a heavy quark mass m_Q. Fitting three mass levels of D mesons by adjusting three parameters included in potentials, other masses of D/B mesons are predicted. All the above calculations are analyzed in 1/m_Q order by order to determine parameters as well as to compare with results of Heavy Quark Effective Theory.

Recent Crystal Barrel(CBAR) experiment of annihilation into three π⁰ on liquid hydrogen target showed resonance like structures of two π⁰ with J^PC = 0⁺⁺ at invariant masses 1365MeV and 1520MeV ^1,2,3. The first one was interpreted as the partner of the (1430) which consists of mainly , while the latter was speculated as a glueball. A Similar resonance was observed in annihilation into π⁺π⁰π⁻ on hydrogen gas target, with quantum numbers 2⁺⁺ and at the mass 1520MeV⁴…

The following sections are included:

• List of Participants